The way the conversion is done here, different sudokus produce different graphs. Besides the regular sudoku graph structure, there are nine additional nodes, each corresponding to one number. They are all connected to each other to ensure they must be different and each one is connected to each cell where the corresponding number is present as a clue from the start. This way, the graph doesn't need any pre-coloring to still encode the sudoku including the given clues.
jkaptur|11 months ago